Consistent estimation of a convex density at the origin

Abstract

Motivated by Hampel’s birds migration problem, Groeneboom, Jongbloed, and Wellner [7] established the asymptotic distribution theory for the nonparametric Least Squares and Maximum Likelihood estimators of a convex and decreasing density, g ₀, at a fixed point t ₀ > 0. However, estimation of the distribution function of the birds’ resting times involves estimation of g′₀ at 0, a boundary point at which the estimators are not consistent.

In this paper, we focus on the Least Squares estimator, $\tilde g_n$. Our goal is to show that consistent estimators of both g ₀(0) and g′₀(0) can be based solely on $\tilde g_n$. Following the idea of Kulikov and Lopuhaä [14] in monotone estimation, we show that it suffices to take $\tilde g_n$(n ^−α) and $\tilde g'_n$(n ^−α), with α ∈ (0, 1/3). We establish their joint asymptotic distributions and show that α = 1/5 should be taken as it yields the fastest rates of convergence.